Stable map

In mathematics, specifically in symplectic topology and algebraic geometry, one can construct the moduli space of stable maps, satisfying specified conditions, from Riemann surfaces into a given symplectic manifold. This moduli space is the essence of the Gromov-Witten invariants. Because the construction is lengthy and difficult, it is carried out here rather than in the Gromov-Witten invariants article itself.

The moduli space of smooth pseudoholomorphic curves

Fix a closed symplectic manifold $X$ with symplectic form $\omega$. Let $g$ and $n$ be natural numbers (including zero) and $A$ a two-dimensional homology class in $X$. Then one may consider the set of pseudoholomorphic curves

$((C,j),f,(x_{1},\ldots ,x_{n}))$

where $(C,j)$ is a closed Riemann surface of genus $g$ with $n$ marked points $x_{1},\ldots ,x_{n}$, and

$f:C\to X$

is a function satisfying, for some choice of $\omega$-tame almost complex structure $J$ and inhomogeneous term $\nu$, the perturbed Cauchy-Riemann equation

${\bar {\partial }}_{j,J}f:={\frac {1}{2}}(df+J\circ df\circ j)=\nu .$

Typically one admits only those $g$ and $n$ that make the punctured Euler characteristic 2 - 2g - n of $C$ negative; then the domain is stable, meaning that there are only finitely many holomorphic automorphisms of $C$ that preserve the marked points.

The operator ${\bar {\partial }}_{j,J}$ is elliptic and thus Fredholm. After significant analytical argument (completing in a suitable Sobolev norm, applying the implicit function theorem and Sard's theorem for Banach manifolds, and using elliptic regularity to recover smoothness) one can show that, for a generic choice of $\omega$-tame $J$ and perturbation $\nu$, the set of $(j,J,\nu )$-holomorphic curves of genus $g$ with $n$ marked points that represent the class $A$ forms a smooth, oriented orbifold

$M_{g,n}(X,A)$

of dimension given by the Atiyah-Singer index theorem,

$d:=\dim _{\mathbb {R} }M_{g,n}(X,A)=2c_{1}^{X}(A)+(\dim _{\mathbb {R} }X-6)(1-g)+2n.$

The stable map compactification

This moduli space of maps is not compact, because a sequence of curves can degenerate to a singular curve, which is not in the moduli space as we've defined it. This happens, for example, when the energy of $f$ (meaning the L²-norm of the derivative) concentrates at some point on the domain. One can capture the energy by rescaling the map around the concentration point. The effect is to attach a sphere, called a bubble, to the original domain at the concentration point and to extend the map across the sphere. The rescaled map may still have energy concentrating at one or more points on the bubble, so one must iterate the process, eventually attaching an entire bubble tree onto the original domain, with the map well-behaved on each smooth component of the new domain.

At this stage the symplectic form $\omega$ enters in a crucial way. The energy is bounded below by the symplectic area $\omega (A)$,

$\omega (A)\leq {\frac {1}{2}}\int |df|^{2},$

with equality if and only if the map is pseudoholomorphic. This inequality implies a bound on how many bubbles one must attach to the domain in order to capture all of the energy.

In the end, the bubbling process (along with other possible degenerations of the domain, if the genus is positive) leads to what is called the stable map compactification

${\bar {M}}_{g,n}(X,A)$

that includes maps on Riemann surfaces with nodal singularities. A sequence of maps in this space converges to a limit map if and only if

1. their domains converge in the Deligne-Mumford moduli space of curves ${\bar {M}}_{g,n}$,
2. they converge uniformly in all derivatives on compact subsets away from the nodes, and
3. the energy concentrating at any point equals the energy in the bubble tree attached at that point in the limit.

The compactified space is again a smooth, oriented orbifold. Due to careful management of the stability condition (which we have glossed over), each map has only finitely many automorphisms. Those with more than one automorphism correspond to points with isotropy in the orbifold.

The Gromov-Witten pseudocycle

The boundary

${\bar {M}}_{g,n}(X,A)\setminus M_{g,n}(X,A)$

of the compactification can be shown to have real codimension at least two. Therefore the image of the space of smooth maps under the evaluation map

$M_{g,n}(X,A)\to {\bar {M}}_{g,n}\times X^{n},$

$((C,j),f,(x_{1},\ldots ,x_{n}))\mapsto ((C,j),f(x_{1}),\ldots ,f(x_{n}))$

is a pseudocycle, and thus represents a rational homology class

$GW_{g,n}^{X,A}\in H_{d}({\bar {M}}_{g,n}\times X^{n},\mathbb {Q} ).$

This homology class is essentially the Gromov-Witten invariant. It is independent of the choice of generic $\omega$-tame $J$ and perturbation $\nu$, and also of the choice of $\omega$ itself, up to isotopy. Thus GW invariants are invariants of symplectic isotopy classes of symplectic manifolds.

References

• Dusa McDuff and Dietmar Salamon, J-Holomorphic Curves and Symplectic Topology, American Mathematical Society colloquium publications, 2004. ISBN 0-8218-3485-1.